Skip to main content
Log in

Size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoshells with double curvature based on nonlocal strain gradient theory

  • Original Article
  • Published:
Engineering with Computers Aims and scope Submit manuscript

Abstract

As initial endeavors, this paper presents an in-depth study to investigate the influence of nanoscale parameters on bending and free vibration responses of functionally graded carbon nanotube-reinforced composite (FG-CNTRC) nanoshells with double curvature. Carbon nanotubes (CNTs) are considered as reinforcements that are distributed across the shell thickness with two different distributions, namely the UD and FG-X. First of all, the mathematical formulas are built on the nonlocal strain gradient theory which, as a critical point of this study, considers both nonlocal and strain gradient parameters simultaneously. Additionally, toward using the Navier solution, the simply supported boundary condition is established to obtain the deflection and natural frequency of FG-CNTRC nanoshells. Furthermore, some specific numerical results are shown and compared with the results reported in the literature. Most importantly, the new findings are given and discussed deeply to show the effect of nanoscale parameters and material property and shape of shells on the deflection and fundamental frequency parameters of the FG-CNTRC nanoshells. From the obtained results, it is shown that the small length-scale has a significant effect on frequencies and deflection of FG-CNTRC nanoshells.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. Harris PJF (2001) Carbon nanotubes and related structures: new materials for the twenty-first century. Cambridge University Press, Cambridge

    Google Scholar 

  2. Wang Q (2009) Atomic transportation via carbon nanotubes. Nano Lett 9:245–249. https://doi.org/10.1021/nl802829z

    Article  Google Scholar 

  3. Arash B, Wang Q, Varadan VK (2014) Mechanical properties of carbon nanotube/polymer composites. Sci Rep 4:6479

    Article  Google Scholar 

  4. Baughman RH, Zakhidov AA, de Heer WA (2002) Carbon nanotubes–the route toward applications. Science 80(297):787–792. https://doi.org/10.1126/science.1060928

    Article  Google Scholar 

  5. Wei BQ, Vajtai R, Ajayan PM (2001) Reliability and current carrying capacity of carbon nanotubes. Appl Phys Lett 79:1172–1174. https://doi.org/10.1063/1.1396632

    Article  Google Scholar 

  6. Thostenson ET, Ren Z, Chou T-W (2001) Advances in the science and technology of carbon nanotubes and their composites: a review. Compos Sci Technol 61:1899–1912. https://doi.org/10.1016/S0266-3538(01)00094-X

    Article  Google Scholar 

  7. Treacy MMJ, Ebbesen TW, Gibson JM (1996) Exceptionally high Young’s modulus observed for individual carbon nanotubes. Nature 381:678–680

    Article  Google Scholar 

  8. Lau AK-T, Hui D (2002) The revolutionary creation of new advanced materials—carbon nanotube composites. Compos Part B Eng 33:263–277. https://doi.org/10.1016/S1359-8368(02)00012-4

    Article  Google Scholar 

  9. Toupin RA (1962) Elastic materials with couple-stresses. Arch Ration Mech Anal 11:385–414. https://doi.org/10.1007/BF00253945

    Article  MathSciNet  MATH  Google Scholar 

  10. Mindlin RD, Eshel NN (1968) On first strain-gradient theories in linear elasticity. Int J Solids Struct 4:109–124. https://doi.org/10.1016/0020-7683(68)90036-X

    Article  MATH  Google Scholar 

  11. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710. https://doi.org/10.1063/1.332803

    Article  Google Scholar 

  12. Eringen AC (1972) Nonlocal polar elastic continua. Int J Eng Sci 10:1–16. https://doi.org/10.1016/0020-7225(72)90070-5

    Article  MathSciNet  MATH  Google Scholar 

  13. Lim CW, Zhang G, Reddy JN (2015) A higher-order nonlocal elasticity and strain gradient theory and its applications in wave propagation. J Mech Phys Solids 78:298–313. https://doi.org/10.1016/j.jmps.2015.02.001

    Article  MathSciNet  MATH  Google Scholar 

  14. Zaera R, Serrano, Fernández-Sáez J (2019) On the consistency of the nonlocal strain gradient elasticity. Int J Eng Sci 138:65–81. https://doi.org/10.1016/j.ijengsci.2019.02.004

    Article  MathSciNet  MATH  Google Scholar 

  15. Hao MJ, Guo XM, Wang Q (2010) Small-scale effect on torsional buckling of multi-walled carbon nanotubes. Eur J Mech 29:49–55. https://doi.org/10.1016/j.euromechsol.2009.05.008

    Article  MathSciNet  MATH  Google Scholar 

  16. Zhang YQ, Liu GR, Han X (2006) Effect of small length scale on elastic buckling of multi-walled carbon nanotubes under radial pressure. Phys Lett A 349:370–376. https://doi.org/10.1016/j.physleta.2005.09.036

    Article  Google Scholar 

  17. Pradhan SC, Reddy GK (2011) Buckling analysis of single walled carbon nanotube on Winkler foundation using nonlocal elasticity theory and DTM. Comput Mater Sci 50:1052–1056. https://doi.org/10.1016/j.commatsci.2010.11.001

    Article  Google Scholar 

  18. Eltaher MA, Emam SA, Mahmoud FF (2012) Free vibration analysis of functionally graded size-dependent nanobeams. Appl Math Comput 218:7406–7420. https://doi.org/10.1016/j.amc.2011.12.090

    Article  MathSciNet  MATH  Google Scholar 

  19. Salari FE, E, (2015) Size-dependent thermo-electrical buckling analysis of functionally graded piezoelectric nanobeams. Smart Mater Struct 24:125007

    Article  Google Scholar 

  20. Ebrahimi F, Barati MR (2017) Buckling analysis of smart size-dependent higher order magneto-electro-thermo-elastic functionally graded nanosize beams. J Mech 33(1):23–33

    Article  Google Scholar 

  21. Şimşek M, Yurtcu HH (2013) Analytical solutions for bending and buckling of functionally graded nanobeams based on the nonlocal Timoshenko beam theory. Compos Struct 97:378–386. https://doi.org/10.1016/j.compstruct.2012.10.038

    Article  Google Scholar 

  22. Thai H-T (2012) A nonlocal beam theory for bending, buckling, and vibration of nanobeams. Int J Eng Sci 52:56–64. https://doi.org/10.1016/j.ijengsci.2011.11.011

    Article  MathSciNet  MATH  Google Scholar 

  23. Ke LL, Xiang Y, Yang J, Kitipornchai S (2009) Nonlinear free vibration of embedded double-walled carbon nanotubes based on nonlocal Timoshenko beam theory. Comput Mater Sci 47:409–417. https://doi.org/10.1016/j.commatsci.2009.09.002

    Article  Google Scholar 

  24. Ebrahimi F, Barati MR (2017) Buckling analysis of nonlocal third-order shear deformable functionally graded piezoelectric nanobeams embedded in elastic medium. J Braz Soc Mech Sci Eng 39:937–952. https://doi.org/10.1007/s40430-016-0551-5

    Article  Google Scholar 

  25. Nejad MZ, Hadi A, Rastgoo A (2016) Buckling analysis of arbitrary two-directional functionally graded Euler-Bernoulli nano-beams based on nonlocal elasticity theory. Int J Eng Sci 103:1–10. https://doi.org/10.1016/j.ijengsci.2016.03.001

    Article  MathSciNet  MATH  Google Scholar 

  26. Wang Q, Liew KM (2007) Application of nonlocal continuum mechanics to static analysis of micro- and nano-structures. Phys Lett Sect A Gen At Solid State Phys 363:236–242. https://doi.org/10.1016/j.physleta.2006.10.093

    Article  Google Scholar 

  27. Farajpour A, Danesh M, Mohammadi M (2011) Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics. Physica E 44:719–727

    Article  Google Scholar 

  28. Jomehzadeh E, Saidi AR (2011) Decoupling the nonlocal elasticity equations for three dimensional vibration analysis of nano-plates. Compos Struct 93:1015–1020. https://doi.org/10.1016/j.compstruct.2010.06.017

    Article  Google Scholar 

  29. Malekzadeh P, Shojaee M (2013) Free vibration of nanoplates based on a nonlocal two-variable refined plate theory. Compos Struct 95:443–452. https://doi.org/10.1016/j.compstruct.2012.07.006

    Article  Google Scholar 

  30. Esawi AMK, Farag MM (2007) Carbon nanotube reinforced composites: potential and current challenges. Mater Des 28:2394–2401

    Article  Google Scholar 

  31. Phung-Van P, Lieu QX, Nguyen-Xuan H, Abdel Wahab M (2017) Size-dependent isogeometric analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Compos Struct 166:120–135. https://doi.org/10.1016/j.compstruct.2017.01.049

    Article  Google Scholar 

  32. Nguyen N-T, Hui D, Lee J, Nguyen-Xuan H (2015) An efficient computational approach for size-dependent analysis of functionally graded nanoplates. Comput Methods Appl Mech Eng 297:191–218. https://doi.org/10.1016/j.cma.2015.07.021

    Article  MathSciNet  MATH  Google Scholar 

  33. Shahriari B, Karamooz Ravari MR, Zeighampour H (2015) Vibration analysis of functionally graded carbon nanotube-reinforced composite nanoplates using Mindlin’s strain gradient theory. Compos Struct 134:1036–1043. https://doi.org/10.1016/j.compstruct.2015.08.118

    Article  Google Scholar 

  34. Gholami R, Darvizeh A, Ansari R, Sadeghi F (2016) Vibration and buckling of first-order shear deformable circular cylindrical micro-/nano-shells based on Mindlin’s strain gradient elasticity theory. Eur J Mech 58:76–88. https://doi.org/10.1016/j.euromechsol.2016.01.014

    Article  MathSciNet  MATH  Google Scholar 

  35. Thai CH, Ferreira AJM, Phung-Van P (2020) A nonlocal strain gradient isogeometric model for free vibration and bending analyses of functionally graded plates. Compos Struct 251:112634. https://doi.org/10.1016/j.compstruct.2020.112634

    Article  Google Scholar 

  36. Phung-Van P, Thai CH (2021) A novel size-dependent nonlocal strain gradient isogeometric model for functionally graded carbon nanotube-reinforced composite nanoplates. Eng Comput. https://doi.org/10.1007/s00366-021-01353-3

    Article  Google Scholar 

  37. Babaei H, Eslami MR (2021) On nonlinear vibration and snap-through buckling of long FG porous cylindrical panels using nonlocal strain gradient theory. Compos Struct 256:113125. https://doi.org/10.1016/j.compstruct.2020.113125

    Article  Google Scholar 

  38. Yuan Y, Zhao X, Zhao Y et al (2021) Dynamic stability of nonlocal strain gradient FGM truncated conical microshells integr ated with magnetostrictive facesheets resting on a nonlinear viscoelastic foundation. Thin-Walled Struct 159:107249. https://doi.org/10.1016/j.tws.2020.107249

    Article  Google Scholar 

  39. Najafi F, Shojaeefard MH, Saeidi Googarchin H (2017) Low-velocity impact response of functionally graded doubly curved panels with Winkler-Pasternak elastic foundation: an analytical approach. Compos Struct 162:351–364. https://doi.org/10.1016/J.COMPSTRUCT.2016.11.094

    Article  Google Scholar 

  40. Shen H, Chen X, Licheng G et al (2015) Nonlinear vibration of FGM doubly curved panel resting on elastic foundations in thermal environments. Aerosp Sci Technol 47:434–446

    Article  Google Scholar 

  41. Tornabene F, Fantuzzi N, Bacciocchi M (2014) Free vibrations of free-form doubly-curved shells made of functionally graded materials using higher-order equivalent single layer theories. Compos Part B Eng 67:490–509

    Article  Google Scholar 

  42. Tornabene F, Liverani A, Caligiana G (2011) FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations. Int J Mech Sci 53:446–470

    Article  Google Scholar 

  43. Chorfi SM, Houmat A (2010) Non-linear free vibration of a functionally graded doubly-curved shallow shell of elliptical plan-form. Compos Struct 92:2573–2581. https://doi.org/10.1016/j.compstruct.2010.02.001

    Article  Google Scholar 

  44. Kar VR, Panda SK (2015) Thermoelastic analysis of functionally graded doubly curved shell panels using nonlinear finite element method. Compos Struct 129:202–212

    Article  Google Scholar 

  45. Alijani F, Amabili M, Karagiozis K, Bakhtiari-Nejad F (2011) Nonlinear vibrations of functionally graded doubly curved shallow shells. J Sound Vib 330:1432–1454. https://doi.org/10.1016/j.jsv.2010.10.003

    Article  Google Scholar 

  46. Oktem AS, Mantari JL, Soares CG (2012) Static response of functionally graded plates and doubly-curved shells based on a higher order shear deformation theory. Eur J Mech 36:163–172. https://doi.org/10.1016/j.euromechsol.2012.03.002

    Article  MATH  Google Scholar 

  47. Duc ND (2013) Nonlinear dynamic response of imperfect eccentrically stiffened FGM double curved shallow shells on elastic foundation. Compos Struct 99:88–96

    Article  Google Scholar 

  48. Bich DH, Van DD, Nam VH (2013) Nonlinear dynamic analysis of eccentrically stiffened imperfect functionally graded doubly curved thin shallow shells. Compos Struct 96:384–395

    Article  Google Scholar 

  49. Thang PT, Nguyen T-T, Lee J (2016) Nonlinear static analysis of thin curved panels with FG coatings under combined axial compression and external pressure. Thin-Walled Struct 107:405–414. https://doi.org/10.1016/j.tws.2016.06.007

    Article  Google Scholar 

  50. Vu-Bac N, Duong TX, Lahmer T et al (2018) A NURBS-based inverse analysis for reconstruction of nonlinear deformations of thin shell structures. Comput Methods Appl Mech Eng 331:427–455. https://doi.org/10.1016/j.cma.2017.09.034

    Article  MathSciNet  MATH  Google Scholar 

  51. Vu-Bac N, Duong TX, Lahmer T et al (2019) A NURBS-based inverse analysis of thermal expansion induced morphing of thin shells. Comput Methods Appl Mech Eng 350:480–510. https://doi.org/10.1016/j.cma.2019.03.011

    Article  MathSciNet  MATH  Google Scholar 

  52. García-Macías E, Rodriguez-Tembleque L, Castro-Triguero R, Sáez A (2017) Buckling analysis of functionally graded carbon nanotube-reinforced curved panels under axial compression and shear. Compos Part B Eng 108:243–256. https://doi.org/10.1016/j.compositesb.2016.10.002

    Article  Google Scholar 

  53. Mehar K, Panda SK, Bui TQ, Mahapatra TR (2017) Nonlinear thermoelastic frequency analysis of functionally graded CNT-reinforced single/doubly curved shallow shell panels by FEM. J Therm Stress 40:899–916. https://doi.org/10.1080/01495739.2017.1318689

    Article  Google Scholar 

  54. Shen H-S, Xiang Y (2016) Postbuckling of pressure-loaded nanotube-reinforced composite doubly curved panels resting on elastic foundations in thermal environments. Int J Mech Sci 107:225–234

    Article  Google Scholar 

  55. Tornabene F, Fantuzzi N, Bacciocchi M, Viola E (2016) Effect of agglomeration on the natural frequencies of functionally graded carbon nanotube-reinforced laminated composite doubly-curved shells. Compos Part B Eng 89:187–218. https://doi.org/10.1016/j.compositesb.2015.11.016

    Article  Google Scholar 

  56. Nguyen TN, Thai CH, Luu A-T et al (2019) NURBS-based postbuckling analysis of functionally graded carbon nanotube-reinforced composite shells. Comput Methods Appl Mech Eng 347:983–1003. https://doi.org/10.1016/j.cma.2019.01.011

    Article  MathSciNet  MATH  Google Scholar 

  57. Nguyen TN, Lee S, Nguyen P-C et al (2020) Geometrically nonlinear postbuckling behavior of imperfect FG-CNTRC shells under axial compression using isogeometric analysis. Eur J Mech 84:104066. https://doi.org/10.1016/j.euromechsol.2020.104066

    Article  MathSciNet  MATH  Google Scholar 

  58. Mahesh V, Harursampath D (2020) Nonlinear vibration of functionally graded magneto-electro-elastic higher order plates reinforced by CNTs using FEM. Eng Comput. https://doi.org/10.1007/s00366-020-01098-5

    Article  Google Scholar 

  59. Daikh AA, Houari MSA, Belarbi MO et al (2021) Analysis of axially temperature-dependent functionally graded carbon nanotube reinforced composite plates. Eng Comput. https://doi.org/10.1007/s00366-021-01413-8

    Article  Google Scholar 

  60. Reddy J (2004) Mechanics of laminated composite plates and shells: theory and analyis. CRC Press, Boca Raton

    Google Scholar 

  61. Amabili M (2008) Nonlinear vibrations and stability of shells and plates. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  62. Thang PT, Tran P, Nguyen-Thoi T (2021) Applying nonlocal strain gradient theory to size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoplates. Appl Math Model 93:775–791. https://doi.org/10.1016/j.apm.2021.01.001

    Article  MathSciNet  MATH  Google Scholar 

  63. Thang PT, Nguyen-Thoi T, Lee D et al (2018) Elastic buckling and free vibration analyses of porous-cellular plates with uniform and non-uniform porosity distributions. Aerosp Sci Technol 79:278–287. https://doi.org/10.1016/J.AST.2018.06.010

    Article  Google Scholar 

  64. Duc ND, Lee J, Nguyen-Thoi T, Thang PT (2017) Static response and free vibration of functionally graded carbon nanotube-reinforced composite rectangular plates resting on Winkler-Pasternak elastic foundations. Aerosp Sci Technol 68:391–402. https://doi.org/10.1016/j.ast.2017.05.032

    Article  Google Scholar 

  65. Şimşek M (2019) Some closed-form solutions for static, buckling, free and forced vibration of functionally graded (FG) nanobeams using nonlocal strain gradient theory. Compos Struct 224:111041. https://doi.org/10.1016/j.compstruct.2019.111041

    Article  Google Scholar 

  66. Thai CH, Ferreira AJM, Nguyen-Xuan H, Phung-Van P (2021) A size dependent meshfree model for functionally graded plates based on the nonlocal strain gradient theory. Compos Struct 272:114169. https://doi.org/10.1016/j.compstruct.2021.114169

    Article  Google Scholar 

  67. Shen JP, Wang PY, Li C, Wang YY (2019) New observations on transverse dynamics of microtubules based on nonlocal strain gradient theory. Compos Struct 225:111036. https://doi.org/10.1016/j.compstruct.2019.111036

    Article  Google Scholar 

  68. Karami B, Janghorban M, Rabczuk T (2020) Dynamics of two-dimensional functionally graded tapered Timoshenko nanobeam in thermal environment using nonlocal strain gradient theory. Compos Part B Eng 182:107622. https://doi.org/10.1016/j.compositesb.2019.107622

    Article  Google Scholar 

  69. Zhu P, Lei ZX, Liew KM (2012) Static and free vibration analyses of carbon nanotube-reinforced composite plates using finite element method with first order shear deformation plate theory. Compos Struct 94:1450–1460. https://doi.org/10.1016/j.compstruct.2011.11.010

    Article  Google Scholar 

  70. Phung-Van P, Abdel-Wahab M, Liew KM et al (2015) Isogeometric analysis of functionally graded carbon nanotube-reinforced composite plates using higher-order shear deformation theory. Compos Struct 123:137–149. https://doi.org/10.1016/j.compstruct.2014.12.021

    Article  Google Scholar 

  71. Pouresmaeeli S, Fazelzadeh SA (2016) Frequency analysis of doubly curved functionally graded carbon nanotube-reinforced composite panels. Acta Mech 227:2765–2794. https://doi.org/10.1007/s00707-016-1647-9

    Article  MathSciNet  Google Scholar 

  72. Van Tham V, Quoc TH, Tu TM (2019) Free vibration analysis of laminated functionally graded carbon nanotube-reinforced composite doubly curved shallow shell panels using a new four-variable refined theory. J Compos Sci 3(4):104

    Article  Google Scholar 

Download references

Acknowledgements

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.02-2019.330.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to T. Nguyen-Thoi.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Thang, P.T., Do, D.T.T., Lee, J. et al. Size-dependent analysis of functionally graded carbon nanotube-reinforced composite nanoshells with double curvature based on nonlocal strain gradient theory. Engineering with Computers 39, 109–128 (2023). https://doi.org/10.1007/s00366-021-01517-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00366-021-01517-1

Keywords

Navigation